curve #4012

y² + xy = x³ + x² − 1262685454892864045086120800x − 3086991656509139192173052611175434290000

a-invariants: [1, 1, 0, -1262685454892864045086120800, -3086991656509139192173052611175434290000]
c-invariants: [60608901834857474164133798425, 2667160791223441695273756024999344223071875]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 46743260634842018618863285423770046153816518554850
discriminant (Δ): 124727636543723156990691670384991642883485394994795766939633076131809640464790250000
Faltings height: 14.6348
naive height: 198.8227
primes of bad reduction: 2, 3, 5, 7, 23, 29, 67, 139, 239, 313, 523, 811, 1193, 1571, 2953, 12113, 14281, 63709, 127691
regulator: 2484293956256037936031635.5762896193234950259147645999820893353841486
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:45:26 UTC
last updated: 2026-10-05 04:45:26 UTC

Witness: 15 independent points log in to add more points →

(1037102328639676635740/9409, 33398917488790106680836471693760/912673)
(-10136958476130015536079815395937680712382850340/1068342058653494076965126440576201, 3131022775622438737198741418680559022870384618192546681246216693956480/34919296783440236155188988830222872571535598626949)
(-104618473051427647296701372813895721511/8322615714954811530702121, 2495080066378339243690221485788818454604989998657083468810/24009867492220776481239205723615509419)
(-2160862062801073060/104329, 4012864344124257436593479040/33698267)
(2127280057346433004863654664/11787857889025, 96168215111873025196059140794596322746748/40471782943994538625)
(333989116917770464131656/319515625, 192906470164489702145178352252206804/5711341796875)
(2883638801196029309159802385/78405782927616, -10338904807862750510995965641984011221695/694259999712293105664)
(-280308451551952001347243095827606912711/8186042659708454348403649, -7215503031630818502734394948437483229908277569023157942/23421300451899721704421238331801218143)
(-813829500393866288432144822170023215/33433341287446652978176, -703044522459237836262441252665029480952165100986372105/6113215033303562062620669948952576)
(51930706535931669395608504812959352889/113319410431916446539769, 373095310391984605766637962802230556315767254278295559434/38146654818054272798864135299855603)
(9187008648112594198773896/242324830225, -7090984724464570537712969741803564164/119288032550709625)
(45754530062314167283921554349241660/338845225050960404401, -9435710354050444120761949037843923875596833845997020/6237377869190007360649163164649)
(19801619048241017878554651064739997385985/317539039578388786254981361, 2268117400700452670857073023700171364347724048593210266168455/5658426647686689944530315510693649090359)
(313869958193281798650754310419369650782684/8254249169834838438111617929, -46715789880734367633841882857484108144040750437128990536457088/749922462201671742931942518452938206357467)
(185047193469173610637807816422589738682065/4113429440399242565587109569, 46562629008380707088122435037867945429199543254530984374506110/263819005010491421832123254548499902750753)

Commentary

Unlisted rational-isogeny neighbor of curve#3647 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 15: 15 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:45:26 UTC · history